252  Mr.  A.  Russell  on  th 
When  V2  is  zero  or  negative  we  see,  since  fi  —f  is  always 
positive,  that  for  a  given  value  of  Vx  — Y2,  Rm  has  its  greatest 
value  when  V2  is  zero  and  has  its  least  value  when  V2=  —  Yx. 
We  shall  now  give  methods  and  formulae  for  calculating 
/and/1?  and  we  shall  also  give  tables  of  their  values. 
8.   Approximate  Formulce  for  the  Maximum  Electric  Intensity 
between  Two  Equal  Spheres. 
Formula  (15)  may  be  written  in  the  form 
1+    l-g    -(l  +  ?3)2? 
f=M 
It  is  easy  to  see  that  the  (?i-f  l)th  term  of  the  series  in  the 
brackets  equals 
l  +  q  +  q2  +  ...  +  q2n  y"  +  l/yn  +  ga-1  +  l/yM-14-  ... 
(1-^  +  q1-. . .  +  g2rt)2 ^  ~  {^+l/^-(^-1  +  l/gn-1)  +  • . .}2' 
By  formula  (3) 
q  +  l/q  =  dja  =  2  +  x\a  =y  (say) , 
and  thus  * 
qn  +  ljqn=yn—nyn-2+  —^^  yn~A - .  .  . 
+  (-1) 
12 
rW(n_r^l)  ...(n-2r  +  r 
«-2r 
/-*=?{ 
We  can  thus  easily  express  /  in  terms  of  y.     Substituting 
and  simplifying  wre  find  that 
y  +  l  V2  +  y-l  ?/3  +  y2-2y-l 
(i/-l)2      (/~i/-l)2      (i/3-^/2-2y  +  l)2 
^yi-y-^  +  l 
(/-y-%2+%+l)2 
/+/-4/-3y2  +  %  +  l 
"(^5-^-%3  +  3y2  +  3^-l)2 
+........}. 
_  Now  y  cannot  be  less  than  2.     Hence  expanding  by  the 
binomial  theorem  and  neglecting  \jy2  and  higher  powers  of 
*  Todhunter's  '  Theory  of  Equations/  3rd  ed.  p.  183. 
